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- W2018852893 abstract "Let I n ={1,2,…, n } and x : I n ↦ R be a map such that ∑ i ∈ I n x i ⩾0. (For any i , its image is denoted by x i .) Let F ={J⊂I n :|J|=k , and ∑ j ∈ J x j ⩾0}. Manickam and Singhi (J. Combin. Theory Ser. A 48 (1) (1988) 91–103) have conjectured that | F |⩾( n−1 k−1 ) whenever n ⩾4 k and showed that the conclusion of the conjecture holds when k divides n . For any two integers r and ℓ let [ r ] ℓ denote the smallest positive integer congruent to r ( mod ℓ) . Bier and Manickam (Southeast Asian Bull. Math. 11 (1) (1987) 61–67) have shown that if k >3 and n ⩾ k ( k −1) k ( k −2) k + k ( k −1) 2 ( k −2)+ k [ n ] k then the conjecture holds. In this note, we give a short proof to show that the conjecture holds when n ⩾2 k +1 e k k k +1 ." @default.
- W2018852893 created "2016-06-24" @default.
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- W2018852893 date "2003-11-01" @default.
- W2018852893 modified "2023-09-29" @default.
- W2018852893 title "On a conjecture of Manickam and Singhi" @default.
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- W2018852893 doi "https://doi.org/10.1016/s0012-365x(03)00192-4" @default.
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